A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as balanced, the resulting tilings have a finite set of prototiles, are quasiperiodic but not periodic, and are self-similar. A necessary and sufficient condition for two balanced tilings to be congruent is provided.

中文翻译：

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来自图有向迭代函数系统的平铺

介绍了一种从图有向迭代函数系统构造欧几里得空间的自引用平铺的新方法，该方法基于我们称为预树的组合结构。在我们称为平衡的特殊情况下，生成的平铺具有有限的原型集，是准周期性的但不是周期性的，并且是自相似的。提供了两个平衡平铺一致的充分必要条件。